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Theorem exmidontri2or 7178
Description: Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
Assertion
Ref Expression
exmidontri2or  |-  (EXMID  <->  A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) )
Distinct variable group:    x, y

Proof of Theorem exmidontri2or
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 exmidontriim 7160 . . 3  |-  (EXMID  ->  A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
2 onelss 4347 . . . . . . . 8  |-  ( y  e.  On  ->  (
x  e.  y  ->  x  C_  y ) )
32adantl 275 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  ->  x  C_  y
) )
4 orc 702 . . . . . . 7  |-  ( x 
C_  y  ->  (
x  C_  y  \/  y  C_  x ) )
53, 4syl6 33 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  ->  ( x  C_  y  \/  y  C_  x ) ) )
6 eqimss 3182 . . . . . . . 8  |-  ( x  =  y  ->  x  C_  y )
76, 4syl 14 . . . . . . 7  |-  ( x  =  y  ->  (
x  C_  y  \/  y  C_  x ) )
87a1i 9 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  =  y  ->  ( x  C_  y  \/  y  C_  x ) ) )
9 onelss 4347 . . . . . . . 8  |-  ( x  e.  On  ->  (
y  e.  x  -> 
y  C_  x )
)
109adantr 274 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( y  e.  x  ->  y  C_  x )
)
11 olc 701 . . . . . . 7  |-  ( y 
C_  x  ->  (
x  C_  y  \/  y  C_  x ) )
1210, 11syl6 33 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( y  e.  x  ->  ( x  C_  y  \/  y  C_  x ) ) )
135, 8, 123jaod 1286 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  ->  (
x  C_  y  \/  y  C_  x ) ) )
1413ralimdva 2524 . . . 4  |-  ( x  e.  On  ->  ( A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x
)  ->  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) ) )
1514ralimia 2518 . . 3  |-  ( A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x )  ->  A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) )
161, 15syl 14 . 2  |-  (EXMID  ->  A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) )
17 ontri2orexmidim 4531 . . . 4  |-  ( A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x )  -> DECID  z  =  { (/)
} )
1817adantr 274 . . 3  |-  ( ( A. x  e.  On  A. y  e.  On  (
x  C_  y  \/  y  C_  x )  /\  z  C_  { (/) } )  -> DECID 
z  =  { (/) } )
1918exmid1dc 4161 . 2  |-  ( A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x )  -> EXMID )
2016, 19impbii 125 1  |-  (EXMID  <->  A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698  DECID wdc 820    \/ w3o 962    = wceq 1335    e. wcel 2128   A.wral 2435    C_ wss 3102   (/)c0 3394   {csn 3560  EXMIDwem 4155   Oncon0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-tr 4063  df-exmid 4156  df-iord 4326  df-on 4328  df-suc 4331
This theorem is referenced by:  onntri52  7179
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